477 research outputs found
The solvability of groups with nilpotent minimal coverings
A covering of a group is a finite set of proper subgroups whose union is the
whole group. A covering is minimal if there is no covering of smaller
cardinality, and it is nilpotent if all its members are nilpotent subgroups. We
complete a proof that every group that has a nilpotent minimal covering is
solvable, starting from the previously known result that a minimal
counterexample is an almost simple finite group
A generalisation of a theorem of Wielandt
In 1974, Helmut Wielandt proved that in a finite group , a subgroup is
subnormal if and only if it is subnormal in every \seq{A,g} for all .
In this paper, we prove that the subnormality of an odd order nilpotent
subgroup of is already guaranteed by a seemingly weaker condition:
is subnormal in if for every conjugacy class of there exists for which is subnormal in \seq{A,c}. We also prove the following
property of finite non-abelian simple groups: if is a subgroup of odd prime
order in a finite almost simple group , then there exists a cyclic
-subgroup of which does not normalise any non-trivial -subgroup
of that is generated by conjugates of~
On the Primary Coverings of Finite Solvable and Symmetric Groups
A primary covering of a finite group is a family of proper subgroups of
whose union contains the set of elements of having order a prime power.
We denote with the smallest size of a primary covering of ,
and call it the primary covering number of . We study this number and
compare it with its analogous , the covering number, for the classes
of groups that are solvable and symmetric
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